Hargreaves heap varoufakis game theory 2nd ed

GAME THEORY, SECOND EDITION Game theory now provides the theoretical underpinning for most areas of economics Moreover, it has spread fast in other disciplines, energised by claims that it represents an opportunity to unify the social sciences, to found a rational theory of society on a common bedrock of methodological individualism The speed of these developments has been remarkable and they have constituted something of a revolution But the technical demands of the subject often discourage the readers most likely to benefit from engaging with it This second edition of Shaun P Hargreaves Heap and Yanis Varoufakis’s classic text strips away the mystique and lets the reader make up his or her own mind It combines the thoroughness of a textbook with the critical edge of the first edition as it: ● ● ● explains clearly all the major concepts (e.g the various forms of Nash’s equilibrium, bargaining solutions), as well as their philosophical bearings (e.g rationality, knowledge, social agency); introduces new, exciting areas of research (e.g psychological, experimental and evolutionary game theory), which it blends carefully with traditional games (e.g the Prisoner’s Dilemma, Hawk–Dove); offers many problems at the end of each chapter, complete with extensive solutions With an uncompromising commitment to intellectual honesty, it seeks out game theory’s strengths and limitations in a bid to draw out their implications for any theory of society which relies exclusively on liberal individualism A new generation of students of game theory will grow to appreciate this superb text whilst fans of the first edition will eagerly devour this long-awaited update Shaun P Hargreaves Heap is Professor of Economics at the University of East Anglia, UK Yanis Varoufakis is Associate Professor of Economics at the University of Athens He is the author of Foundations of Economics, also published by Routledge GAME THEORY, SECOND EDITION A critical text Shaun P Hargreaves Heap and Yanis Varoufakis First edition published in 1995 This revised edition published 2004 by Routledge 11 New Fetter Lane, London EC4P 4EE Simultaneously published in the USA and Canada by Routledge 29 West 35th Street, New York, NY 10001 Routledge is an imprint of the Taylor & Francis Group This edition published in the Taylor & Francis e-Library, 2004 © 2004 Shaun P Hargreaves Heap and Yanis Varoufakis All rights reserved No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalog record for this book has been requested ISBN 0-203-48929-2 Master e-book ISBN ISBN 0-203-56960-1 (Adobe eReader Format) ISBN 0–415–25094–3 (hbk) ISBN 0–415–25095–1 (pbk) CONTENTS List of boxes Preface xi xiii OVERVIEW 1.1 Introduction 1.1.1 Why study game theory? 1.1.2 What is game theory? 1.1.3 Why this book? 1.1.4 Why a second book? 1.1.5 The rest of this chapter 1.2 The assumptions of game theory 1.2.1 Individual action is instrumentally rational 1.2.2 Common knowledge of rationality (CKR) 27 1.2.3 Common priors 28 1.2.4 Action within the rules of the games 31 1.3 Liberal individualism, the state and game theory 1.3.1 Methodological individualism 33 1.3.2 Game theory’s contribution to liberal individualism 35 1.4 A guide to the rest of the book 1.4.1 Three classic games: Hawk–Dove, Co-ordination and the Prisoner’s Dilemma 36 1.4.2 Chapter-by-chapter guide 38 1.5 Conclusion THE ELEMENTS OF GAME THEORY 2.1 Introduction 2.2 The representation of strategies, games and information sets 2.2.1 Pure and mixed strategies 44 2.2.2 The normal form, the extensive form and the information set 45 2.3 Dominance reasoning 2.4 Rationalisable beliefs and actions 2.4.1 The successive elimination of strategically inferior moves 52 2.4.2 Rationalisable strategies and their connection with Nash’s equilibrium 56 2.5 Nash equilibrium 2.5.1 John Nash’s beautiful idea 58 v 1 33 36 40 41 41 44 47 52 58 CONTENTS 2.5.2 Consistently aligned beliefs, the hidden Principle of Rational Determinacy and the Harsanyi–Aumann doctrine 60 2.5.3 Some objections to Nash: Part I 61 2.6 Nash equilibrium in mixed strategies 2.6.1 The scope and derivation of Nash equilibria in mixed strategies 68 2.6.2 The reliance of NEMS on CAB and the Harsanyi doctrine 73 2.6.3 Aumann’s defence of CAB and NEMS 75 2.7 Conclusion Problems BATTLING INDETERMINACY: REFINEMENTS OF NASH’S EQUILIBRIUM IN STATIC AND DYNAMIC GAMES 3.1 Introduction 3.2 The stability of Nash equilibria 3.2.1 Trembling hand perfect Nash equilibria 81 3.2.2 Harsanyi’s Bayesian Nash equilibria and his defence of NEMS 85 3.3 Dynamic games 3.3.1 Extensive form and backward induction 90 3.3.2 Subgame perfection, Nash and CKR 92 3.3.3 Sequential equilibria 96 3.3.4 Bayesian learning, sequential equilibrium and the importance of reputation 99 3.3.5 Signalling equilibria 103 3.4 Further refinements 3.4.1 Proper equilibria 106 3.4.2 Forward induction 108 3.5 Some logical objections to Nash, Part II 3.5.1 A critique of subgame perfection 111 3.5.2 A negative rejoinder (based on the Harsanyi–Aumann doctrine) 114 3.5.3 A positive rejoinder (based on sequential equilibrium) 115 3.5.4 Summary: out-of-equilibrium beliefs, patterned trembles and consistency 117 3.6 Conclusion 3.6.1 The status of Nash and Nash refinements 118 3.6.2 In defence of Nash 119 3.6.3 Why has game theory been attracted ‘so uncritically’ to Nash? 122 Problems BARGAINING GAMES: RATIONAL AGREEMENTS, BARGAINING POWER AND THE SOCIAL CONTRACT 4.1 Introduction 4.2 Credible and incredible talk in simple bargaining games 4.3 John Nash’s generic bargaining problem and his axiomatic solution 4.3.1 The bargaining problem 135 4.3.2 Nash’s solution – an example 137 vi 68 78 79 80 80 81 90 106 111 118 125 127 127 131 135 CONTENTS 4.3.3 Nash’s solution as an equilibrium of fear 140 4.3.4 Nash’s axiomatic account 146 4.3.5 Do the axioms apply? 148 4.4 Ariel Rubinstein and the bargaining process: the return of Nash backward induction 4.4.1 Rubinstein’s solution to the bargaining problem 150 4.4.2 A proof of Rubinstein’s theorem 152 4.4.3 The (trembling hand) defence of Rubinstein’s solution 160 4.4.4 A final word on Nash, trembling hands and Rubinstein’s bargaining solution 163 4.5 Justice in political and moral philosophy 4.5.1 The negative result and the opening to Rawls and Nozick 165 4.5.2 Procedures and outcomes (or ‘means’ and ends) and axiomatic bargaining theory 168 4.6 Conclusion Problems THE PRISONER’S DILEMMA: THE RIDDLE OF CO-OPERATION AND ITS IMPLICATIONS FOR COLLECTIVE AGENCY 5.1 Introduction: the state and the game that popularised game theory 5.2 Examples of hidden Prisoner’s Dilemmas and free riders in social life 5.3 Some evidence on how people play the Prisoner’s Dilemma and free rider games 5.4 Explaining co-operation 5.4.1 Kant and morality: is it rational to defect? 185 5.4.2 Altruism 186 5.4.3 Inequality aversion 187 5.4.4 Choosing a co-operative disposition instrumentally 189 5.5 Conditional co-operation in repeated Prisoner’s Dilemmas 5.5.1 Tit-for-Tat in Axelrod’s tournament 191 5.5.2 Tit-for-Tat as a Nash equilibrium strategy when the horizon is unknown 192 5.5.3 Spontaneous public good provision 194 5.5.4 The Folk Theorem and Indeterminacy in indefinitely repeated games 196 5.5.5 Does a finite horizon wreck co-operation? The theory and the evidence 202 5.6 Conclusion: co-operation and the State in Liberal theory 5.6.1 Rational co-operation? 205 5.6.2 The debate in Liberal political theory 206 5.6.3 The limits of the Prisoner’s Dilemma 209 Problems EVOLUTIONARY GAMES: EVOLUTION, GAMES AND SOCIAL THEORY 6.1 Introduction 6.1.1 The origins of Evolutionary Game Theory 212 6.1.2 Evolutionary stability and equilibrium: an introduction 214 vii 150 164 170 171 172 172 175 180 185 191 205 209 211 211 CONTENTS 6.2 Symmetrical evolution in homogeneous populations 6.2.1 Static games 220 6.2.2 Dynamic games 223 6.3 Evolution in heterogeneous populations 6.3.1 Asymmetrical (or two-dimensional) evolution and the demise of Nash equilibria in mixed strategies 227 6.3.2 Does Evolutionary Game Theory apply to humans as well as it does to birds, ants, etc.? An experiment with two-dimensional evolution in the Hawk–Dove game 232 6.3.3 Multi-dimensional evolution and the conflict of conventions 236 6.3.4 The origin of conventions and the challenge to methodological individualism 241 6.3.5 The politics of mutations: conventions, inequality and revolt 245 6.3.6 Discriminatory conventions: a brief synopsis 247 6.4 Social evolution: power, morality and history 6.4.1 Social versus natural selection 248 6.4.2 Conventions as covert social power 251 6.4.3 The evolution of predictions into moral beliefs: Hume on morality 252 6.4.4 Gender, class and functionalism 255 6.4.5 The evolution of predictions into ideology: Marx against morality 258 6.5 Conclusion Problems PSYCHOLOGICAL GAMES: DEMOLISHING THE DIVIDE BETWEEN MOTIVES AND BELIEFS 7.1 Introduction 7.2 Different types of ‘other regarding’ motives 7.2.1 The ‘other’ regarding motives of Homo Economicus 268 7.2.2 Beliefs as predictions and as motives 269 7.3 The power of normative beliefs 7.3.1 Fairness equilibria 275 7.3.2 Computing fairness equilibria 281 7.3.3 An assessment of Rabin 283 7.3.4 An alternative formulation linking entitlements to intentions 285 7.3.5 Team thinking 289 7.4 Psychology and evolution 7.4.1 On the origins of normative beliefs: an adaptation to experience 292 7.4.2 On the origins of normative beliefs: the resentment-aversion versus the subversion-proclivity hypotheses 293 7.5 Conclusion: shared praxes, shared meanings Problems Postscript Answers to problems 220 227 248 264 266 267 267 268 275 292 299 301 302 304 viii CONTENTS Notes Bibliography Name index Subject index 334 348 359 362 ix BIBLIOGRAPHY Myerson, R (1991) Game Theory: Analysis of Conflict Cambridge, MA: Cambridge University Press Nash, J (1950) ‘The bargaining problem’, Econometrica, 18, 155–62 Nash, J (1951) ‘Non-cooperative games’, Annals of Mathematics, 54, 286–95 Nash, J (1953) ‘Two person cooperative games’, Econometrica, 21, 128–40 Nasar, S (1998) A Beautiful Mind New York: Simon 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on game theory 1; Harsanyi–Aumann doctrine 28, 29, 60, 66, 114–15 Axelrod, R 183, 191, 202–3 Axtell, R 258 Babcock, L 293 Bacharach, M 40 Bayes, T 22, 23 Becker, G 255 Bentham, J 10 Bergin, J 247 Bergstrom, T 250 Bernheim, D 56 Binmore, K 3, 150, 152 Brams, S Buchanan, J 208 Burke, E 207 Carr-Saunders, A 248 Chomsky, N 247 Coleman, A 180 Comte, A 207 Condorcet, J A 207, 251 Cournot, A A 307, 308, 309 Darwin, C 212, 213, 217, 258–9 Davies, D 204 Dawes, R 182 Dawkins, R 221, 248, 249, 250 Descartes, R 207 Diamond, P 218 DiMaggio, J 235 Dixit, A 3, Edwards, W 248 Ellsberg, D 26 Elster, J 2, 3, 124–5, 257 Engels, F 129, 259 Epstein, J M 258 Eshel, I 251 Faludi, S 256 Ferguson, A 207 Festinger, L 20 Fisher, R 217 Fodor, J 249 Foster, D 247 Frank, R 181, 182 Frey, B 188 Friedman, D 245 Friedman, M 212 Fudenberg, D Gauthier, D 189, 190, 191 Geanakoplos, J 40 Giddens, A 32 Goeree, J 62, 133, 163 Gould, S J 77, 217 Gramsci, A 252 Guth, W 162 Habermas, J 11 Hammond, R 258 Hardin, R 183, 185 Hargreaves Heap, S 121 Harsanyi, J.: Bayesian Nash equilibrium 85–9, 92, 279–80; defence of CAB 78; Harsanyi–Aumann doctrine 28, 29, 60, 114–15; incomplete information 31; Nobel Prize 92 Hart, S 1, 252 Hayek von, F 36, 207, 208, 254 Hegel, G W F 19, 30, 31, 40, 120, 124 Hobbes, T 34, 35, 36, 38, 127, 129, 174–5, 206–7 Holt, C 62, 133, 163, 204 Howard, J V 153 Howard, N 190 Hume, D.: agency 3, 21, 119; conventions 120, 123, 293; morality 253, 259; political development 207; reason 10, 30–1, 119, 253; social interaction 34; trust 176 359 NAME INDEX Isaac, R 205 Kahneman, D 16 Kalai, E 149, 169 Kandori, M 247 Kant, I.: beliefs 64, 76; categorical imperative 19, 20, 64, 186; economics 124; morality 185–6, 190, 253, 259; reason 30–1, 120 Keynes, J M 64, 68 Kohlberg, E 109, 120 Kreps, D 3, 63, 97, 103 Ledyard, J 204 Lewis, D 241 Lipman, B L 247 Lukes, S 251, 252, 254 Lyotard, J.-F 122–3 McKelvey, R 203 MacKinnon, C 129–30 McPherson, C B 260–1 Mailath, G 226 Malthus, T R 212 Marshall, A 212 Marx, K 19–20, 33, 129, 178–9, 258–64, 265, 300 Medea 299 Mehta, J 243 Mertens, J.-F 109, 120 Meyer, D 202 Mill, J S 10, 257, 294 Miller, J 181 Morgenstern, O 2–3, 61 Myerson, R 2, 3, 88, 107, 108, 184, 202–3 Nalebuff, B Nash, J.: backward induction 95; bargaining problem 130, 140–1, 143, 146–7, 151–2, 315–16; commitment to 122, 124–5; Nash equilibrium 38, 58, 59, 63; Nobel Prize 92; population-statistical interpretation 212–13; rationalisable strategies 58; uncertainty 85 Nelson, R 213 Nietzsche, F W 192 Nozick, R 36, 165, 168, 254 Ockenfels, A 187 Olson, M 195 O’Neill, O 30 Orbell, J 182 Osborne, M Palfrey, T 203 Pateman, C 129 Pierce, D 56 Pinker, S 295 Plato 259 Polanyi, K 300 Popper, K 214 Prometheus 298 Rabin, M 40, 268, 275–8, 281–4, 286, 288–9, 291, 299 Rapport, A 191 Rasmussen, E Rawls, J 165–8 Robinson, J 179 Roemer, J 169–70 Rousseau, J.-J 34, 67–8, 200, 207, 209, 216 Rubinstein, A 4, 150–64, 315 Runciman, W 251 Samuelson, L 213, 226 Savage, L 26 Schelling, T 3, 243, 257–8 Schumpeter, J 212 Schuster, P 221 Selten, R 81, 91, 92, 161, 187 Sen, A 17–18 Sigmund, K 221 Skeath, S Smith, A 40, 178, 187, 207, 271, 272, 273, 300 Smorodinsky, M 149, 169 Socrates 28, 29, 259 Soros, G 66 Spartacus 252, 298 Spence, M 103 Spöhn, W 56 Stark, O 250 Stinchcombe, A 209 Sugden, R.: altruism 187; bargaining games 163–4; conventions 253–4, 293; Evolutionary Game Theory 225; Prisoner’s Dilemma 194; psychological games 40, 76, 271, 292–7, 299; salience 242 Sweezy, P 212 Thaler, R 182 Thatcher, M 254 Tirole, J Titmuss, R 188 Tucker, A 172, 173, 207 Tullock, G 40, 208 Turnbull, C 185 Tversky, A 16 Ulysses 190 van Huyck, J 200 van Parij, P 257 360 NAME INDEX Varoufakis, Y 122 Visser, M 244 von Neumann, J 2–3, 43, 61 Wagner, R 208 Walker, J 205 Walras, L 201 Weber, M 11, 19 Wilson, R 97, 103 Winter, S 213 Wittgenstein, L 6, 20, 32, 33, 40, 120, 124, 242, 273, 299 Wolinsky, A 150 Young, H P 247 Zeuthen, F 314, 315 361 SUBJECT INDEX action: communicative 11; rational 6, 7, 11, 15, 27, 119; rules of the game 31–3; structure relationship 32, 265, 302; value rational 11; see also collective action agency 3, 18, 21, 64, 213, 264–5 Allais paradox 16–17, 148 altruism 186–7, 251, 272 ambiguity 26 analogy 242 antibiotics 180 Arrow Impossibility Theorem 208 asocial individuals 33–4 auctions 49 Austrian School: critique of socialism 21, 208; entrepreneurs 64; institution-creation 36 axiomatic approach 130, 131, 146–50, 169 backward induction 91–2, 93, 95, 96, 99, 118; bargaining games 130, 132, 151, 152–3, 155–6, 159; common knowledge of rationality 213; Evolutionary Game Theory 226; Prisoner’s Dilemma 203–4, 318–19, 321; problem solutions 318–19; sequential equilibrium 98, 117, 119; see also forward induction; Nash backward induction Bank of England 65 bargaining games 38, 127–71; credible/incredible talk 131–5; justice in moral/political philosophy 164–70; problem solutions 314–17 bargaining problem 128–30; axiomatic approach 130, 131, 146–50, 169; Nash’s solution 38, 130, 135–50, 152–3, 164–5, 168, 170, 315–17; Rubinstein’s solution 150–64 baseball 32–3, 70, 77 Battle-of-the-Sexes game 69, 134, 222, 224, 229, 231–2 Bayesian consistency 118, 119 Bayesian Nash equilibrium 87–9, 279–80, 310–11 Bayes’s Rule 22–3, 24, 27; common priors 28; problem solutions 312, 313; sequential equilibrium 99, 101, 102, 110, 323; zero-probability events 113 A Beautiful Mind 38, 40, 130 beliefs: bargaining games 137; Bayesian consistency 118, 119; Bayes’s Rule 99; Evolutionary Game Theory 241, 252; fairness equilibria 275–9, 284; group co-operation 197; Harsanyi–Aumann doctrine 60, 66; Kant 64; moral 253, 254, 259, 260, 263, 264, 265; motivation 267–8, 275; Nash equilibrium in mixed strategies 73–4, 75–6; normative 275–98; out-of-equilibrium 107, 113, 115, 118, 119; psychological games 39, 269–75, 279–80, 284–5; rational 6, 15; rationalisable strategies 56–8; second order 267–8, 270, 275, 280, 289, 290; sequential equilibrium 98–9, 323–4; sexist 105–6; signalling behaviour 104; source of 21–5; see also consistent alignment of beliefs; expectations best reply strategies: Evolutionary Game Theory 220; Nash equilibrium 41, 42, 53, 56, 61–2; Prisoner’s Dilemma 193–4, 225; psychological games 285, 286, 289; sequential equilibrium 97, 98, 107; tragedy of the commons 50 bidding strategies 49 biology 213, 217, 230, 232, 241–2, 245, 247; see also Darwinism; Evolutionary Game Theory blood donation 188–9 bluffing: irrationality 101, 102, 115–16, 312; poker 70; Prisoner’s Dilemma 319, 321, 322, 323, 324, 325; tremble distinction 170 CAB see consistent alignment of beliefs capitalism 178–9, 212, 261–3, 300 cardinal utility 11–13 catchment area 223 categorical imperative 19, 20, 21, 64, 66, 186 Centipede game 117, 131, 203–4, 295, 303; Evolutionary Game Theory 225; subgame perfect Nash equilibrium 92, 96, 99, 100, 112, 115 cheap talk 131, 132, 133 chess 63 362 SUBJECT INDEX Chicken game see Hawk–Dove game CKR see common knowledge of rationality class: conflict 178–9; Marxism 262–3, 264, 300; stratification 255–7 cognitive dissonance 20–1, 254, 262 collective action 36, 175, 176, 209; discriminatory conventions 246, 247; Evolutionary Game Theory 265; Marxism 259 collective interest 67–8 commitment 67–8 Common Assurance game see Stag-Hunt game common good 253–4, 259 common knowledge of rationality (CKR) 6, 27–8, 30; bargaining games 151, 170; economists 124, 125; forward induction 108, 111; mixed strategies 74; Nash equilibrium 42, 60, 62, 78, 118–19; order of 48–9, 54, 55–6; out-of-equilibrium behaviour 113, 114, 115; Prisoner’s Dilemma 203; problem solutions 305–6, 308, 309, 318–19, 325; psychological games 280; rationalisable strategies 57; relaxation of 99, 100, 103, 116, 325; sequence of moves 90; subgame perfect Nash equilibrium 93, 94, 95, 132, 213; successive elimination of inferior moves 52–4, 55–6; trembles 83; uncertainty 99, 100 common priors 6, 28–31, 78, 118; see also consistent alignment of beliefs the ‘commons’ 50, 128 communication 131, 134, 205; see also cheap talk; promises; threats communicative action 11 completeness conflict 128, 130, 134–5, 137, 141–2, 170, 314 consistent alignment of beliefs (CAB) 28, 30, 44, 63, 78, 122; backward induction 95; bargaining games 134, 135, 145–6, 155, 159, 170; definition 58–9; economists 124, 125; Harsanyi–Aumann doctrine 28–9, 60, 114; hidden assumption 155; Nash equilibrium in mixed strategies 73, 74, 75, 76; Nash equilibrium refinements 89, 118, 119; psychological games 279–80; sequential equilibrium 323–4; static games 118; see also beliefs; common priors constructivism 208, 265, 266 consumer choice 21 continuity 8, 12 contractarianism 183 conventions: bargaining games 148, 149, 150; capitalist 262, 263; discriminatory 233–4, 236–7, 240, 246–7, 254, 259–60, 296; Evolutionary Game Theory 233, 234, 236–41, 242, 246, 247, 251–2; fairness 164; gender/race 255–6; individualism 34; institution-creation 34, 36; justice 253, 289; moral 253–4; Nash equilibrium 120; normative beliefs 293; social evolution 248, 259, 260, 261, 262, 263; social power 251–2; subversive-proclivity hypothesis 296–7 co-operation: bargaining problem 128, 130; conditional 183, 188, 191–205, 269; economists 181, 182; Evolutionary Game Theory 225–6, 234, 235, 236, 249–51, 292; explaining 185–91; fairness equilibria 275, 276; group 182, 195, 197; instrumental rationality 189–91; libertarianism 208–9; norms 185; Prisoner’s Dilemma 175, 181–3, 191–205, 206, 208, 223, 225–6, 317–24; rational 205–6 co-operative game theory 39, 131 Co-ordination game 37, 69, 222, 223, 224, 290–1 corruption 179, 258, 330–3 Cournot equilibrium 307–9 credibility 132, 133, 141–2, 146, 161, 174 cricket 32–3, 70 custom 120, 253 Darwinism: Evolutionary Game Theory 214, 217, 248; Marx 258; psychological 295; see also biology democracy 34 deviant behaviour 112–14, 116, 118; see also out-of-equilibrium behaviour dialogue 28–9 difference principle 166 disarmament 177 discount rate 152, 154, 159 discriminatory conventions 233–4, 236–7, 240, 246–7, 254, 259–60, 296 dispositions 39, 189–91 domestic labour 177 dominance reasoning 38, 47–51, 93, 181, 182; see also strategies, dominant drama 298–9 dynamic games 90–106, 117, 118–19; bargaining 130, 152–3; Evolutionary Game Theory 213–14, 223–6; opportunity to condition behaviour 192; see also extensive form economics: capitalist ideology 262–3; central planning 21; co-operative norms 185; evolutionary ideas 212–13; expectations 23; functional explanations 124–5; influence on Prisoner’s Dilemma behaviour 181, 182; instrumental rationality 123–4, 299; Keynesian 64, 68, 218; philosophical controversies 4; recession 68; see also Austrian School; neoclassical economics EE see evolutionary equilibrium EFA see Equilibrium Fear Agreements 363 SUBJECT INDEX efficiency 136, 137, 169, 200, 201 Ellsberg paradox 25–6 empiricism 123, 207 employers’ beliefs 103–6 enforcement mechanisms 131, 174–5 Enlightenment 122–3, 207, 253 entitlement theory 165 equality 165, 168 equilibrium: Bayesian Nash 87–9, 279–80, 310–11; Bayesian perfect 102; Cournot 307–9; evolutionary 215–26, 228–34, 241–2, 244, 246–7, 295, 326–7, 333; fairness 275–9, 281–3, 286, 288–9, 291, 330; gender/racial differentiation 257; normative expectations equilibrium 293–4, 296, 298; proper 106–8, 117; psychological games 274, 275; reflective 169; selection of 111, 200–1, 215, 225, 241–3, 257, 265; sequential 96–9, 101–3, 107, 109–10, 115–17, 312–13, 323–5; signalling 103–5, 110; ‘solution’ synonymity 41; trembling hand 81–5, 109–10, 161–2; Walrasian 201–2; see also multiple equilibria; Nash equilibrium; Nash equilibrium in mixed strategies; out-of-equilibrium behaviour; subgame perfect Nash equilibrium Equilibrium Fear Agreements (EFA) 142, 143–6 equity 268–9; see also fairness ERM see Exchange Rate Mechanism ESS see evolutionarily stable strategy ethical preferences 188, 189, 205–6 eudaimonia 259 European Exchange Rate Mechanism (ERM) 65 EvGT see Evolutionary Game Theory evolutionarily stable strategy (ESS) 219–20, 221 evolutionary equilibrium (EE) 215–26, 228–34, 241–2, 244, 246–7, 295, 326–7, 333 Evolutionary Game Theory (EvGT) 5–6, 39, 208–9, 211–66, 267, 302; corruption 332–3; dynamic games 213–14, 223–6; heterogeneous populations 227–47; homogeneous populations 220–6, 232, 233; origins of 212–14; problem solutions 325–9, 332–3; psychological games 292, 295, 296, 297, 298; social evolution 236, 242, 248–64; stability 215–16, 219–20, 228, 230, 246; static games 220–3 Exchange Rate Mechanism (ERM) 65 expectations: Bayesian Nash equilibrium 87; common knowledge of rationality 27–8; common prior 28, 29, 31; Harsanyi–Aumann doctrine 60; instrumental rationality 21–7; moral 253; normative expectations equilibrium 293–4, 296, 298; power of prophecy 68; psychological games 267, 270–1, 273–5; second order beliefs 267, 270; subversive-proclivity hypothesis 294, 296; uncertainty 64; see also beliefs expected utility 9–13, 14, 15–17, 26, 32, 71, 167–8 extensive form 45–7, 90–1, 95, 108, 118; see also dynamic games extractive power 261, 262, 263, 264 fairness 162–3, 164, 169; advocacy 293; equilibria 275–9, 281–3, 286, 288–9, 291, 330; psychological games 268–9, 294; resentment-aversion hypothesis 294 farming 260, 261, 262, 263 fashion 297 fear see Equilibrium Fear Agreements feedback mechanism 257 feminism 129 First World War 183–5, 206 Folk Theorem 192, 202, 206, 208, 247 forward induction 108–11, 117, 119, 120 free rider problem 176–85, 186, 194–6, 205, 206–7, 209 Freudianism 242 functional explanations 124–5, 257 game definition gender relations 255–6, 257; see also sexism genes 221, 248, 249, 295 global warming 176 grand narratives 122–3 Great War 183–5, 206 group co-operation 182, 195, 197, 205 group-interest 248 Harsanyi–Aumann doctrine 28–9, 31, 60, 63–4, 66, 118; bargaining games 163, 164; equilibrium selection 201; Nash backward induction 114–15; Nash equilibrium in mixed strategies 76; symmetrical games 72; veil of ignorance 167 Harsanyi doctrine 60, 73, 76, 78 Hawk–Dove (Chicken) game 36–7, 69, 110, 130; beliefs 271; conventions 237, 238–9, 242, 246, 252; eating dinner 244; entitlements 286; Evolutionary Game Theory 214–16, 218–19, 221–2, 224, 229–36, 241, 292; fairness equilibria 277–8, 280, 281–3, 289; property rights 255; psychological game theory 288, 289, 292, 293; social evolution 260, 263 haystack models 248–50 Hegelianism 121 hegemony 252 hidden assumption 155–6, 157, 159 Hide and Seek game 69, 224, 227, 329 historical time 47, 77 364 SUBJECT INDEX Homo Economicus 7–8, 21, 123, 268–9, 271, 272, 300 honour 15–17, 297 hunting-gathering 260–1 ideology 259, 262–3 IIA see Independence of Irrelevant Alternatives incredible threats 132, 133, 151, 159 independence axiom 12, 17 Independence of Irrelevant Alternatives (IIA) 146, 147, 148–50, 316–17 Independence of Utility Calibrations 146, 147 indeterminacy 39, 58, 80, 117, 121, 301; bargaining games 131, 138; Evolutionary Game Theory 211, 213, 229, 231–2, 233, 247; instrumental rationality 264, 265; Nash equilibrium in mixed strategies 68, 70, 71, 72, 74, 78; postmodernism 123; Prisoner’s Dilemma 175, 192, 199, 202; psychological games 285; subversive logic 295; trembles 245; see also rational determinacy indifference 51, 71, 72–3 indifference curves individualism: liberal 3, 7, 35–6, 302, 303; limits of 3; methodological 33–5, 238, 243–4, 301, 302–3 induction see backward induction; forward induction; Nash backward induction inequality: aversion to 187–9; gender/race 255, 257; social 165, 166 inference 63, 64; Bayes’s Rule 24, 27; common priors 28, 30; Nash equilibrium 66–7 information: asymmetrical 86, 89, 135; Bayes’s Rule 22–3; complete 31–2, 85, 279; exchange of 29; imperfect 176; incomplete 85–6, 88, 89; instrumental rationality 23–5; symmetry 73–4; see also veil of ignorance information set: definition of 46; forward induction 109; Harsanyi doctrine 60, 66; sequential equilibrium 97–8; subgame definition 93–4 institutions 34, 35–6; enforcement mechanisms 131; impartiality 165–6; instrumental rationality 184 instrumental rationality 6, 7–8, 10–11, 15–26, 33, 40; bargaining 164–5; classic games 36; convention relationship 120; co-operation 189–91; dominant strategies 48; dynamic games 117; economics 123–4, 299; expected utility 167; fairness 164; indeterminacy 264, 265; Mill 257; pessimism 184; Prisoner’s Dilemma 204, 205, 300, 319, 320–4; psychological games 284–5; rejection of 121; social processes 121, 122; zero/first-order of CKR 48–9 intentions 82–3, 285–9, 290 Intransigent Right 207–8, 254 invisible hand 178 Iron Rule of Selfishness 250 irrationality: common knowledge of 117; feigning 113–14; miners’ strike 135; probability of 99, 100, 101, 102, 103, 116 iterated dominance see Successive Elimination of Strictly Dominated Strategies justice 135, 164–70; conventions 253, 289; gender/racial inequalities 255; Nozick 165; Rawls 165–8; social 254 Keynesian economics 64, 68, 218 kindness 268, 275–7, 284, 285, 287, 288 labour market 103–6 language games 33, 273, 299 learning 213, 257, 264 leximin rule 169–70 liberal individualism 3, 7, 35–6, 302, 303 liberalism: spontaneous order/social constructivism debate 265, 266; the State 127–8, 129, 164–5, 175, 205, 206–9, 266 libertarianism 207–9 live-and-let-live 183–5, 206 logical time 47, 77, 90 luck 235 marginal per capita return (MPCR) 204, 205 Marienbad game 91, 92, 93 markets 201–2 Marxism 258–64; functional explanations 257; preferences 121; rational choice 11; the State 129 maximin principle (Rawlsian) 166–7, 168 maximin strategy 43, 201 means–ends conflicts 168–9 meta-narratives 122–3 methodological individualism 33–5, 238, 243–4, 301, 302–3 minimax theorem 43 mixed strategies 44–5, 68–77, 280; see also Nash equilibrium in mixed strategies modernity 11, 122–3 monopoly game 88 monotonicity 149, 169 morality: Aristotle 252–3; common knowledge of rationality 280; conventions 253–4; co-operative behaviour 185–6, 187–9, 190; Hume 253; judgements 272, 273; Kantian 185–6, 253; luck 235; Marxism 259, 263; moral beliefs 253, 254, 259, 260, 263, 264, 265; moral philosophy 130, 164–70; preference satisfaction 17; psychological games 270 365 SUBJECT INDEX motivation: beliefs 267–8, 275; Hume 3; instrumental rationality 190; moral 186, 187–9, 267; norms 299, 300; ‘other’ regarding motives 268–75; Smith 300 MPCR see marginal per capita return multi-dimensional evolution 236–41 multiple equilibria 5, 34, 66–7, 164, 206; bargaining games 130; Folk Theorem 202; indeterminacy 138; pure strategies 68; repeated games 39 mutation mechanism 221, 228, 245–6, 247, 248, 332 Nash backward induction 93, 95, 96, 99, 108, 109–16; bargaining games 132, 151, 152–3, 155–6, 159; Evolutionary Game Theory 226; Prisoner’s Dilemma 203–4, 318–19; problem solutions 318–19; sequential equilibrium 98; see also backward induction Nash equilibrium 5, 38, 41, 42–4, 58–68, 78–9; abandonment of 121–2; bargaining games 130, 133, 134, 137–8, 161, 164, 170; Bayesian 87–9, 279–80, 310–11; beauty of 59; consistent alignment of beliefs 28; definition 42; discovery of 2; dominance reasoning 49, 51, 52; Evolutionary Game Theory 214, 216–19, 225–6, 228, 230–2, 233, 245; fairness equilibria 276, 278–9, 283, 288–9; Folk Theorem 202; forward induction 109, 111; Hawk–Dove game 214; Humean turn 119–20; Kantian move 120; multiple 39, 202, 208; pooling 104, 106; popular culture 40; ‘population-statistical interpretation’ 212–13; Prisoner’s Dilemma 172, 175, 192–4, 196, 198–201, 206; problem solutions 304–6, 308–9, 311, 318, 330; public good provision 183; rationalisable strategies 56; Refinement Project 38, 78–9, 80–126, 164, 245, 267; reliance upon 122–5; separating 104, 105, 106; sequential equilibria 96–9, 101–3, 110; social evolution 260; stability 80, 81; successive elimination of inferior moves 52–6; symmetrical 72; trembling hand perfect 81–5, 109–10, 161–2; see also subgame perfect Nash equilibrium Nash equilibrium in mixed strategies (NEMS) 68–77, 78, 85–9, 134–5, 153; beliefs 279; definition 74; Evolutionary Game Theory 214–16, 218–20, 221–4, 228–32, 233, 240; forward induction 111; problem solutions 304–6, 309, 311, 325, 326, 327–8 Nash’s axioms 130, 146–50 Nash solution to the bargaining problem 38, 130, 135–50, 152–3, 164–5, 168, 170, 315–17 natural selection 212, 214, 217, 248, 249, 257 NEMS see Nash equilibrium in mixed strategies neoclassical economics 3, 4, 123, 124; beliefs 25; instrumental rationality 10–11; utility maximisation 9, 167; see also economics neo-liberalism 21 New Right 36 Nim game 91–2, 93 NME see normative expectations equilibrium non-cooperative game theory 39, 130, 131 normal form 45, 46–7, 91, 108 normative expectations equilibrium (NME) 293–4, 296, 298 norms 17, 18, 258, 264, 300–1; capitalist society 262; fairness 275, 291; of justice 253; live-and-let-live 183–5, 206; motivations 299, 300; psychological games 6, 285; social evolution 260; subversive-proclivity hypothesis 297–8 oligopoly 306–9 one-shot games 47, 191, 192, 205 optimism 68 ordinal utility out-of-equilibrium behaviour 107, 113, 114–15, 117, 118, 119; bargaining games 160, 162, 163; see also deviant behaviour; irrationality; trembles Pareto Optimality 146, 147, 148 parlour games 91 path-dependence 53, 55 patriarchy 129, 177 perturbed games 81, 82, 89 pessimism 43, 68, 184, 197 phenotypes 214, 230, 232 philosophy 10, 18; see also political philosophy poker 70 political philosophy: justice 164–70; liberal individualism 3, 7, 35–6 political rationalism 208–9 politics: corruption 179, 330–3; Evolutionary Game Theory 246, 247; Intransigent Right 207–8, 254; power 251; see also State pooling Nash equilibrium 104, 106 popular culture 40 postmodernity 122–3 power: extractive 261, 262, 263, 264; social 251–2, 260 preferences 3, 299; cardinal utility 11–12, 13; cognitive dissonance 20; ethical 188, 189, 205–6; expected utility 9–11, 12; Humean view 10, 119, 120; indeterminacy 285; instrumental rationality 7–8, 17; ordering 7, 8, 9; ‘other’ regarding motives 268, 269; psychological games 39; team 291, 292; time 154, 157–8, 159 primitive accumulation 259–60 366 SUBJECT INDEX Prisoner’s Dilemma 37–8, 39, 172–210, 299–300, 303; altruism 186; conditional co-operation 191–205; co-operative dispositions 189–91; corruption 330, 331, 332; entitlements 286–7; Evolutionary Game Theory 223, 224, 225; examples of 175–80; experiments 180–1; fairness equilibria 275–6, 281–3, 289; Kantian morality 186; limits of 209; mixed strategies 69; ‘other’ regarding motives 268–9; problem solutions 317–24, 330, 331, 332; psychological game theory 274, 275, 288, 289, 292; strictly dominant strategies 48; team thinking 291; see also free rider problem probability: Allais paradox 16–17; Bayesian Nash equilibrium 87; Bayes’s Rule 22–3, 24; expected utility 9–11, 12, 16–17, 26; incomplete information 86; mixed strategies 70, 71, 73, 75; of opponent’s irrationality 99, 100, 101, 102, 103, 116; Prisoner’s Dilemma 193, 194, 196; proper equilibrium 107; trembling hand perfect Nash equilibrium 83 prominence 242, 243, 244 promises 132, 174, 182 proper equilibrium 106–8, 117 property rights 127–8, 237, 244, 254, 255–6 prophecy 68, 197 psychological Darwinism 295 psychological games 6, 39, 267–301, 302; entitlements/intentions link 285–9; and evolution 292–8; fairness equilibria 275–9, 281–3, 286, 288–9, 291; ‘other’ regarding motives 268–75; problem solutions 329–33; team thinking 289–92 psychology: bargaining games 160; evolutionary 5; social 121 public goods: altruism 186; free rider problem 177, 194–6; generic provision problem 182, 183; marginal per capita return 204–5; Rousseau 67; State provision 209; see also the ‘commons’ pure strategies 44, 45, 56, 59, 62; Evolutionary Game Theory 214, 216, 231, 232, 233; Nash equilibrium in mixed strategies 68–9, 70, 71, 72–3, 75, 78, 88; trembling hand perfect Nash equilibrium 81, 82, 83, 84 pygmies 185 QWERTY 217–18, 296 racial differentiation 255–6, 257 racism 258 RAH see resentment-aversion hypothesis Rand Corporation 180–1 randomisation 69–70, 71, 72, 73, 74, 82, 102–3 rational action 6, 7, 11, 15, 27, 119 rational choice theory 3, 290; instrumental rationality 11, 12; subversion of 6; weakness of rational determinacy 60, 63–4, 119, 122, 164; see also indeterminacy rationalisability 56–8, 61, 121, 137, 151, 306, 308 rationalisation 20–1, 48 rationality: common priors 28; defining 21; Enlightenment 122; Evolutionary Game Theory 213–14, 234, 241; game theory predictions 5; Harsanyi–Aumann doctrine 60, 63–4, 66, 78; Humean 10, 21; independence axioms 148, 149; Kantian 19, 20, 21, 39, 64, 120, 186; Nash equilibrium 59, 60, 61, 62–3, 78, 164; out-of-equilibrium behaviour 117; Prisoner’s Dilemma 37–8, 39; sequential 106; trembles 115; uncertainty 99, 100; see also common knowledge of rationality; instrumental rationality; irrationality reason: Harsanyi–Aumann doctrine 63–4; Hegel 19, 40; Hume 3, 10, 30, 31, 119, 120, 253; instrumental 10–11, 18–19, 20, 121–2; Kantian 19, 30, 64; reflecting on 30, 31; see also dominance reasoning; instrumental rationality; rationality reciprocation: fairness equilibria 275, 276, 277, 283–4; Prisoner’s Dilemma 183, 185, 205, 206 reflective equilibrium 169 reflexivity 8, 30 repeated games 39, 84; constrained maximisation 191; Prisoner’s Dilemma 175, 191–205, 206, 225 replicator dynamics 221–2, 224, 226, 227–8, 234, 326–7, 332 reputation 39, 102, 103, 191, 195–6, 312, 313–14 resentment 294, 296 resentment-aversion hypothesis (RAH) 294, 295, 296 resource distribution 169–70 the Right 36, 207, 254 risk: aversion 14, 135, 136, 138–9, 140, 152, 154; cardinal utility 13; of conflict 141, 142; neutrality 14, 135–6, 138; uncertainty distinction 26, 64 rules 6, 7, 31–3, 63–4; evaluative 271; Kantian 120; leximin 169–70; moral 273; Nash’s solution to the bargaining problem 146–7; shared 32, 33; Wittgenstein 6, 32, 33, 299 salience 242, 243, 244, 247 security principle 200, 201 selection mechanism 221, 245, 248 self-interest 17, 167, 178, 181, 204, 263, 264 selfish gene theory 248, 249 367 SUBJECT INDEX Selfishness, Iron Rule of 250 self-punishment 110, 111 separating Nash equilibrium 104, 105, 106 sequence of moves 90–1 sequential equilibria 96–9, 101–3, 107, 109–10, 115–17, 312–13, 323–5 SESDS see Successive Elimination of Strictly Dominated Strategies sexism 105–6, 148, 255, 256; see also gender relations shared practices 299 Short Centipede game 92, 96, 99, 100, 112, 131 signalling behaviour 103–5, 110, 111 singletons 93, 94 social class see class social constructivism 208, 265, 266 social context 238, 242, 244 social contract 34, 207 social evolution 236, 242, 248–64 social interaction 31, 33, 209; classic games 36; conventions 34, 242; team thinking 290, 291 socialism 21, 208 social justice 254 social power 251–2, 260 social psychology 121 social science 3, 4, 6, 40, 121, 209, 211 social selection 248 social theory 2, 40; action–structure relationship 32; Evolutionary Game Theory 246, 249, 251; rational choice 11 sociology 123, 124, 209 solidarity 21 SPH see subversive-proclivity hypothesis SPNE see subgame perfect Nash equilibrium spontaneity 15, 34 spontaneous order 35–6, 39, 208, 251–2, 265, 266 stability: Evolutionary Game Theory 215–16, 219–20, 228, 230, 246; Nash equilibrium 80, 81 Stag-Hunt game 67–8, 69, 90–1, 329; equity 269; Evolutionary Game Theory 215–16, 217, 218–20, 221, 222–3, 224; fairness equilibria 283; Prisoner’s Dilemma transformation into 188, 200; Walrasian equilibrium 201 State 34, 35–6, 38, 266; bargaining problem 128, 129, 164, 165; enforcement 174–5, 177; Evolutionary Game Theory 211; feminism 129–30; Marxism 129; Prisoner’s Dilemma 205, 206–9; property rights 127–8; repeated games 39; social constructivism 265; spontaneous order 35, 36, 251–2; see also politics state of nature 33–4, 35, 36, 127, 129, 174, 177 strategies: backward induction 91, 92; bargaining problem 137, 151, 160, 161–2; Bayesian Nash equilibrium 87; evolutionarily stable 219–20; Evolutionary Game Theory 213, 215, 219–20, 221–3, 225–9, 233–5; fairness equilibria 276, 277–8, 280, 281; Folk Theorem 202; long-term 199–200; maximin 43, 201; mixed 44–5, 68–77, 280; Nash equilibrium 38, 59–68; rationalisable 56–8, 61, 121, 137, 151, 306, 308; sequential equilibrium 97–8, 101–2; strictly dominant 47–8, 50, 86, 174, 177; strictly dominated 47–8, 52–3, 54–6, 81, 223, 306, 308; subgame perfect Nash equilibrium 92, 93, 94, 95, 97; trembling hand perfect Nash equilibrium 81–5; weakly dominant 51; weakly dominated 51, 53, 55, 81, 83–4, 85; see also best reply strategies; Nash equilibrium in mixed strategies; pure strategies strategy profile 97, 98 structure 32, 33–4, 265, 302 subgame perfect Nash equilibrium (SPNE) 91, 92–5, 96–7, 115–16, 117, 213; backward induction 96, 99, 100, 111–12; bargaining games 130–1, 132, 133, 151–5, 159–60, 162–4; Prisoner’s Dilemma 192, 203, 204, 318–19, 325; ultimatum game 162–3, 226 subgames: definition 93–4; subgame perfection 97–8, 106, 118, 132, 160–1, 295 subjectivity 25, 26, 75 subversive-proclivity hypothesis (SPH) 294, 295–8, 299 Successive Elimination of Strictly Dominated Strategies (SESDS) 54–6, 306, 308, 309 symmetry: distribution of resources 169; Evolutionary Game Theory 221; fairness equilibria 276; informational 73–4; Nash equilibrium in mixed strategies 72, 78; Nash’s bargaining solution 146, 147, 148 sympathy 40, 271, 272, 273 taxes 20, 177, 208 team thinking 275, 289–92 technology 260, 261, 262 tennis 77 terrorism 168 threats 132–4, 141–2, 146, 151, 159, 161, 170 time 47, 77, 90 time preferences 154, 157–8, 159 Tit-for-Tat: co-operation 191–6, 198–9, 200, 203, 206, 209, 225; problem solutions 317, 318, 319, 320–4 torture 168 Tosca’s dilemma 174 trade unions 128, 149, 178, 179, 259 tragedy of the commons 50 transitivity tree diagram see extensive form 368 SUBJECT INDEX trembles 38, 107, 109–10, 114–15, 117, 118–19; bargaining games 160–4; irrationality 101, 113; mutation comparison 245; Nash equilibrium 81–5, 89; Prisoner’s Dilemma 198, 204; see also irrationality; out-of-equilibrium behaviour trust 176 truthfulness 49 ultimatum game 162–3, 226 uncertainty: bargaining games 154; Bayes’s Rule 99; cardinal utility 11, 13; complete information 31; Ellsberg paradox 26; expected utility 9, 11, 12, 15; Harsanyi 85, 86; mixed strategies 44, 45; risk distinction 26, 64; sequential equilibrium 116; trembles 85, 89 unemployment 218 universalisability 120 UPF see Utility Possibility Frontier utilitarianism 9, 10, 123, 167, 168, 187 utility: bargaining problem 135–7, 138–9, 140, 143, 144–5, 148; cardinal 11–13; co-operation 187; expected 9–13, 14, 15–17, 26, 32, 71, 167–8; Independence of Utility Calibrations 146, 147; information acquisition 23–5; maximin principle 166, 167; maximisation 8–9, 11, 12, 13, 167–8; ordinal 9; psychological 269–70, 275–6, 278–9, 281–3, 284, 288; utilitarianism 10 Utility Possibility Frontier (UPF) 136–7, 138–9, 140–3, 145, 147, 153 value rational action 11 veil of ignorance 166–7, 168 Walrasian equilibrium 201–2 zero-sum games 43 369 ... Yanis Varoufakis First edition published in 1995 This revised edition published 2004 by Routledge 11 New Fetter Lane, London EC4P 4EE Simultaneously published in the USA and Canada by Routledge... opposed to game theory for self-interested reasons but, rather, from practising game theorists (see Mailath, 1998 and Samuelson, 2002) Our book’s point in 1995 was that game theory is best studied... studying game theory 1.1.2 What is game theory? Game theory really begins with the publication of The Theory of Games and Economic Behaviour by John von Neumann and Oskar Morgenstern (first published